Purpose The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface. Design/methodology/approach The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time. Findings The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems. Originality/value For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

The unsmooth boundary of a cavity for the stage of filling glass fiber-reinforced polymer melt affects the flow property and the shape of the free interface’s shape, as a result, it will affect greatly the process, the degree and the location of the front interface, and the weld lines’ property and product quality as well. Here, the two-scale fractal theory is adopted to figure out the basic property of the unsmooth boundary, and governing equations are established in a fractal space. The numerical results can provide insight into the filling process and reveal the main factors affecting the shape of the free interface. The fractal interface is illustrated by the least-square method in different fractal spaces based on numerical simulation results. The fractal cavity sheds a new light on controlling the interface.